Child of constraint and moving rig in Object mode
I've always found that parenting gear such as clothes, etc. to an armature bone would be sufficient enough to handle animating the character, armour, belts, etc. and also moving the character around in Object Mode all the same. 

After testing the "Child Of" constraint on a simple mesh with a single armature and a "belt" (reshaped Sphere) I noticed the problem you described - the object always moves against the direction of the bone/armature.

All in all, I think it may be worth considering the option of directly parenting (Ctrl + P shortcut) the belt and other objects to the bone that controls them (i.e. via "Set Parent To" -> "Bone" option) and then other necessary parents (e.g. bone(s) to empty parent through "Set Parent To" option) and see if this method doesn't work better for you.